Character table for point group D8h

D8h E 2C8 2(C8)3 2C4 C2 (z) 4C2' 4C2'' i 2(S8)3 2S8 2S4 h 4v 4d
linear functions,
rotations
quadratic
functions
cubic
functions
A1g +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 - x2+y2, z2 -
A2g +1 +1 +1 +1 +1 -1 -1 +1 +1 +1 +1 +1 -1 -1 Rz - -
B1g +1 -1 -1 +1 +1 +1 -1 +1 -1 -1 +1 +1 +1 -1 - - -
B2g +1 -1 -1 +1 +1 -1 +1 +1 -1 -1 +1 +1 -1 +1 - - -
E1g +2 +(2)½ -(2)½ 0 -2 0 0 +2 +(2)½ -(2)½ 0 -2 0 0 (Rx, Ry) (xz, yz) -
E2g +2 0 0 -2 +2 0 0 +2 0 0 -2 +2 0 0 - (x2-y2, xy) -
E3g +2 -(2)½ +(2)½ 0 -2 0 0 +2 -(2)½ +(2)½ 0 -2 0 0 - - -
A1u +1 +1 +1 +1 +1 +1 +1 -1 -1 -1 -1 -1 -1 -1 - - -
A2u +1 +1 +1 +1 +1 -1 -1 -1 -1 -1 -1 -1 +1 +1 z - z3, z(x2+y2)
B1u +1 -1 -1 +1 +1 +1 -1 -1 +1 +1 -1 -1 -1 +1 - - -
B2u +1 -1 -1 +1 +1 -1 +1 -1 +1 +1 -1 -1 +1 -1 - - -
E1u +2 +(2)½ -(2)½ 0 -2 0 0 -2 -(2)½ +(2)½ 0 +2 0 0 (x, y) - (xz2, yz2) [x(x2+y2), y(x2+y2)]
E2u +2 0 0 -2 +2 0 0 -2 0 0 +2 -2 0 0 - - [xyz, z(x2-y2)]
E3u +2 -(2)½ +(2)½ 0 -2 0 0 -2 +(2)½ -(2)½ 0 +2 0 0 - - [y(3x2-y2), x(x2-3y2)]


Additional information

Number of symmetry elements h = 32
Number of irreducible representations n = 14
Abelian group no
Number of subgroups36
Number of distinct subgroups20
Subgroups
(Number of different orientations)
Cs (3) , Ci , C2 (3) , C4 , C8 , D2 (2) , D4 (2) , D8 , C2v (4) , C4v (2) , C8v , C2h (3) , C4h , C8h , D2h (2) , D4h (2) , D2d (2) , D4d (2) , S4 , S8
Optical Isomerism (Chirality) no


Reduction formula for point group D8h

Type of representation

general 3N vib

E 2C8 2(C8)3 2C4 C2 (z) 4C2' 4C2'' i 2(S8)3 2S8 2S4 h 4v 4d




Multipoles

dipole (p) A2u+E1u
quadrupole (d) A1g+E1g+E2g
octopole (f) A2u+E1u+E2u+E3u
hexadecapole (g) A1g+B1g+B2g+E1g+E2g+E3g
32-pole (h) A2u+B1u+B2u+E1u+E2u+2E3u
64-pole (i) A1g+B1g+B2g+E1g+2E2g+2E3g
128-pole (j) A2u+B1u+B2u+2E1u+2E2u+2E3u
256-pole(k) 2A1g+A2g+B1g+B2g+2E1g+2E2g+2E3g
512-pole (l) A1u+2A2u+B1u+B2u+3E1u+2E2u+2E3u

First nonvanishing multipole: quadrupole

Literature



Character tables for chemically important point groups Computational Laboratory for Analysis, Modeling and Visualization Jacobs University Bremen